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Efendiyev, M.A. ; Sengul, T.* ; Tiryakioglu, B.*

Two approaches to instability analysis of the viscous Burgers' equation.

Discret. Contin. Dyn. Syst.-Ser. S 17, 1621-1638 (2023)
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The 1D Burger's equation with Dirichlet boundary conditions exhibits a first transition from the trivial steady state to a sinusoidal patterned steady state as the parameter lambda which controls the linear term exceeds 1. The main goal of this paper is to present two different approaches regarding the transition of this patterned steady state. We believe that these approaches can be extended to study the dynamics of more interesting models. As a first approach, we consider an external forcing on the equation which supports a sinusoidal solution as a stable steady state which loses its stability at a critical threshold. We use the method of continued fractions to rigorously analyze the associated linear problem. In particular, we find that the system exhibits a mixed type transition with two distinct basins for initial conditions one of which leads to a local steady state and the other leaves a small neighborhood of the origin. As a second approach, we consider the dynamics on the center-unstable manifold of the first two modes of the unforced system. In this approach, the secondary transition produces two branches of steady state solutions. On one of these branches there is another transition which indicates a symmetry breaking phenomena.
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Publication type Article: Journal article
Document type Scientific Article
Keywords Burger's equation; secondary transition; center manifold; continued fraction; Hopf-bifurcation; Dynamics; Waves
Language english
Publication Year 2023
HGF-reported in Year 2023
ISSN (print) / ISBN 1937-1632
e-ISSN 1937-1179
Journal Discrete and Continuous Dynamical Systems - Series S
Quellenangaben Volume: 17, Issue: 4, Pages: 1621-1638 Article Number: , Supplement: ,
Publisher American Institute of Mathematical Sciences (AIMS)
Publishing Place Po Box 2604, Springfield, Mo 65801-2604, United States
Reviewing status Peer reviewed
Institute(s) Institute of Computational Biology (ICB)
POF-Topic(s) 30205 - Bioengineering and Digital Health
Research field(s) Enabling and Novel Technologies
PSP Element(s) G-503800-001
DOI 10.3934/dcdss.2023202
WOS ID 001101379700001
Scopus ID 85190274853
Erfassungsdatum 2024-01-10
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